166 XIII INTEGRATION 8. (a) Let (/*)i^^« be a finite orthonormal sequence in ^c(X, /*) Show that there exists a constant c independent of n such that, for every finite sequence (tf*)io<n of complex numbers, we have sup j (Suppose first that n 2r. For each 7 g /?, let .s/ = Z a*/>" Split UP tn*s sum mto at most r + 1 partial sums corresponding to the dyadic expansion of the number ;, so r that Sj = 2 pq where, for each q, either pq = 0 or else pq is of the form q = Q s «*/». with 0 <: h < 2r~*. Using Cauchy's inequality |^|2 <^ r X l^l2, show that «=o Finally, if 2r <; n< 2r+1, take the <2k to be 0 for k > «.) (b) Let (w(n))nzi be an increasing sequence of integers >0, and tending to +cx>. Let (/)«£! be an orthonormal sequence in ^c(^j M)- If *ne sequence («)>! is such that X W(w)l^nl2< +°°> and if (wfc) is a sequence of integers such that k <^ w(rik) <k+l9 then the sequence of partial sums snk = J] «,/// converges almost everywhere in X. (If /e ^c(X, ju-) has the fln as coefficients (6.5.2) with respect to the system (/), evaluate N2(/ snk) rnk say, and remark that the series with general term r Jk converges, using Abel's partial summation technique by writing rjfc = (k + 1 k)r*k (Section 13.21, Problem 6).) (c) Suppose that the sequence (an) is such that]£ |^n|2(log n)2 < +00. Show that the n series ^ anfn converges almost everywhere ("Rademacher-MenchofT theorem")- n (First deduce from (b) that the sequence (s2k) converges almost everywhere. Then use (a) to show that, if Sfc = sup \sn s2k\, the series with general term Si du, 2fc4«<2*+1 J converges.) 9. Suppose that the measure /x is bounded, so that L2(X, /A) <= LJ(X, p). Suppose also that jLi(X)= 1. Let U be a continuous endomorphism of L1 with norm ^1 (a "con- traction ") whose restriction to L2 is also a contraction (i.e., N2(t/ /) ^ N2(7) in L2). 1 n~1 Let P be the endomorphism of L2 which is the strong limit of - T Uk (Section 12.1 5, n *=o Problem 12), so that P = PU = UP =P2. Show that P can be extended by con- tinuity to;an endomorphism of L1 satisfying the same relations and that P-/«lim -ZV-/ n -» oo n k =s o for all/e L1 (use the fact that L2 is dense in L1 and that N^/) ^ N2(/).) with general term anfn(x)